Mathematical models of systems can be used in a wide variety of disciplines. These mathematical models can be used to predict how a system will react in different environments and can be used to improve the system. One approach to mathematical modeling may be known as “white box” modeling. In this approach, the laws governing a system are represented mathematically. For example, for a physical system, the law of gravity may be represented in the model. The outputs of the system may be determined by applying the mathematical model to the inputs. Sometimes, however, system behavior is too complex to be easily modeled using the “white box” approach.
Another approach to mathematical modeling may be known as system identification (SysId). System identification may include mathematical modeling of systems from experimental data and may be applied in many engineering areas, e.g., communications, mechanical engineering, or geophysical engineering for purposes such as spectral analysis, fault detection, adaptive filtering, linear prediction, and pattern recognition. SysId techniques may also find application in fields such as biology, environmental sciences and econometrics.
In SysId, a system can be treated as a black box, i.e., inputs can be applied to an unknown system, and a corresponding (noise corrupted) output signal from that system can be observed. These input and output signals may then be used, together with a mathematical model of the system, to estimate the model parameters that best characterize the system. When the system has N input channels, traditionally one may repeat the experiment N times with a different input channel selected to be active for each repetition.